Subgroups — #30664019

Order 16768.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₃₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22553065	C₂	1	1	yes	maximal, characteristic
3	#500×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₁₃₁⋊C₈)	—	2	1	no	—
6	C₄⋊Q₈×C₁₃₁	—	2	1	no	—
7	#107×C₁₃₁	—	2	1	no	—
8	#107×C₁₃₁	C₄	1	2	yes	—
9	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₁⋊C₈	—	4	1	no	—
11	C₁₃₁⋊C₁₆	—	4	1	no	—
12	C₂×Q₈×C₁₃₁	—	4	1	no	—
13	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₁₃₁	—	2	2	no	—
16	C₄⋊C₄×C₁₃₁	—	4	1	no	—
17	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
18	C₁₃₁⋊C₈	—	8	1	no	—
19	Q₈×C₁₃₁	—	8	2	no	—
20	C₂×C₄×C₁₃₁	—	4	2	no	—
21	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₃₁	—	2	1	no	—
25	C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
26	C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₁	#102	1	1	yes	characteristic
28	C₄×C₁₃₁	—	2	1	no	—
29	C₄×C₁₃₁	—	8	2	no	—
30	C₄×C₁₃₁	#101	1	1	yes	characteristic
31	C₄×C₁₃₁	#103	1	1	yes	characteristic
32	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
33	C₄×C₁₃₁	Q₃₂	1	1	yes	characteristic
34	C₂×C₁₃₁	#332	1	1	yes	characteristic
35	C₂×C₁₃₁	#358	1	1	yes	characteristic
36	C₂×C₁₃₁	#361	1	1	yes	characteristic
37	C₁₃₁	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	131	1	no	maximal, Sylow
39	#500	D₁₃₁	1	1	yes	characteristic
40	C₄⋊C₁₆	—	131	1	no	—
41	#335	—	131	1	no	—
42	C₄⋊Q₈	—	2	1	no	—
43	C₂×C₁₆	—	262	1	no	—
44	#107	—	2	1	no	—
45	#107	Dic₁₃₁	1	2	yes	—
46	C₄⋊C₈	—	262	1	no	—
47	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
48	C₂×Q₈	—	4	1	no	—
49	C₂×C₈	—	524	1	no	—
50	C₂×C₈	C₂×Dic₁₃₁	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
55	C₁₆	—	524	1	no	—
56	Q₈	—	8	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
59	C₂×C₄	#14523498	1	1	yes	characteristic
60	C₂×C₄	#14523501	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	1048	1	no	—
63	C₈	#14523500	1	1	yes	characteristic
64	C₈	#14523503	1	1	yes	characteristic
65	C₂²	#16862167	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	8	2	no	—
68	C₄	#16862165	1	1	yes	characteristic
69	C₄	#16862170	1	1	yes	characteristic
70	C₄	#16862189	1	1	yes	characteristic
71	C₄	#16862192	1	1	yes	characteristic
72	C₂	#22553058	1	1	yes	characteristic
73	C₂	#22553116	1	1	yes	characteristic
74	C₂	#22553125	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862167
commutator	C₂×C₈×C₁₃₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₃₁
Fitting	#500×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#102
cosocle	C₄×C₈×C₁₃₁	C₂²
hypercenter	#500	D₁₃₁
hypocenter	C₁₃₁	#1365
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
37. C₁₃₁	#1365	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊(C₁₃₁⋊C₁₆)	2
#22553065	2
#500×C₁₃₁	2
#1365	131